Minimal semi-flat-cotorsion replacements and cosupport
arXiv:1907.04671
Abstract
Over a commutative noetherian ring of finite Krull dimension, we show that every complex of flat cotorsion -modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every -complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over . Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.
28 pages. Final version to appear in Journal of Algebra. We have made a number of minor revisions, including modifications to Lemma 1.1, a new Lemma 3.8, and a corrected Proposition A.11